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Suppose that Y is a continuous random variable with a cumulative distribution function given by F(y) and a probability density function f(y). We are often interested in conditional probabilities of the form P(Y < y | Y > c) for some constant c. Show that for y > c, P(Y < y | Y > c) = [F(y) - F(c)] / [1 - F(c)] Note that this function has all the properties of a distribution function. (b) Refer to (a). If the length of life, Y, for a battery has a Weibull distribution with a = 2 and B = 3 (with measurements in years), find the probability that the battery will last less than four years given that the battery is still running after two years. Recall that the cumulative distribution for the Weibull distribution is given in Question 5(a). (c) Since P(Y < y | Y > c) does possess the properties of a distribution function, its derivative will have the properties of a probability density function. As a result, show that the expected value of Y given that Y is greater than c is given by: E(Y | Y > c) = ∫[y * f(y)]dy / [1 - F(c)] Refer to (c). If Y, the lifetime of an electronic component, has an exponential distribution with a mean of 100 hours, find the expected value of Y given that this component has already been in use for 50 hours.
Submitted by Tonya T. Sep. 07, 2021 07:21 p.m.
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5 comments
Gary R.
March 9, 2023
You're the real deal! Keep up the awesome work!
Katrina R.
April 25, 2023
big thanks ameer your breakdown really cleared things up for me! keep those explanations coming
Troy A.
August 17, 2023
The answer is way too jumbled! Can we get a more organized explanation for each part? It's hard to follow the steps and calculations with the current response.??
Tyler A.
August 27, 2023
This explanation is a bit confusing Can we get a clearer walkthrough of the solution
Katherine C.
September 18, 2023
this explanation could use some clarification can someone explain how to find the conditional probability py y y > c using the given weibull distribution parameters and then simplify the numerical example in part (b